Multivariate Statistical Data Analysis

The basis of the following data analysis is a row data matrix with 119 samples and 12 features (11 element concentrations and the mass of the dried total-day portions). The feature Cu has to be eliminated because some of the measurements are missing. The feature dried mass must also be deleted, because it is an extensive variable. The original values were then autoscaled. 

Firstly PCA was performed according to the mathematical principles described in Section 5.4. The loadings of the three most important principal components are presented in Tab. 10-2. 

Principal component 1 is highly loaded by elements introduced by salting of the food. The second principal component is highly loaded by Fe and P. The third principal component, which is highly loaded by Cd and Pb, hints at anthropogenically caused entry of these heavy metals. 

MVDA with an a priori model of two classes is used to classify multidimensional differences between communal and individual feeding. The significant separation strengths of the individual features are 1.036 for Na, 0.0928 for K, and 0.0426 for Mg. 

The other features do not separate the classes significantly. The optimum separation set  

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The multivariate statistical data analysis, using principal component analysis (PCA), of this historical data revealed three main contamination profiles. A first contamination profile was identified as mostly loaded with PAHs. A samples group which includes sampling sites R1 (Ebro river in Miranda de Ebro, La Rioja), T3 (Zadorra river in Villodas, Alava) and T9 (Arga river in Puente la Reina, Navarra), all located in the upper Ebro river basin and close to Pamplona and Vitoria cities,... 

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

Similar data evaluation problems exist in other scientific fields and can also be treated by multivariate statistical data analysis, for instance, in economics (econometrics), sociology, psychology (psychometrics), medicine, biology (chemotaxonomy),... 

Conventional microbiological identification of isolates from patients can normally be obtained with a total turnaround time of 48-96 h. Ibelings et al. [106] and Maquelin et al. [46] developed alternatively a Raman spectroscopic approach for the identification of clinically relevant Candida species from smears and microcolonies in peritonitis patients taking at least overnight (smears) or about 6h (microcolonies). Hereby, a prediction accuracy of 90% was obtained for Raman spectroscopy in combination with multivariate statistical data analysis. 

Chemical Differentiation and Multivariate Statistical Data Analysis... 

Chemometrics The application of multivariate statistics, data analysis, predictive modeling, and data mining to problems within chemistry. 

The first reported use of PTR-MS for food, rather than for drink, perception was by Gasperi et al. [19]. This was also the first food study that compared classical sensory analysis with VOC headspace analysis by PTR-MS. The foods chosen for investigation were seven varieties of Italian mozzarella cheese, for which measurements were made while they were held at a temperature of 36°C. By using a multivariate statistical data analysis approach. 

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

Graphical and statistical data analysis will be carried out at various scales (regional, States/Northern Territory, and National). Non-parametric univariate and multivariate analysis along with the production of geochemical maps will be carried out. 

Gnanadesikan, R. (1977). Method for statistical data analysis of multivariate observations. Wiley, New York. 

In Chapter 2, we approach multivariate data analysis. This chapter will be helpful for getting familiar with the matrix notation used throughout the book. The art of statistical data analysis starts with an appropriate data preprocessing, and Section 2.2 mentions some basic transformation methods. The multivariate data information is contained in the covariance and distance matrix, respectively. Therefore, Sections... 

Gnanadesikan, R., Methods for Statistical Data Analysis of Multivariate Observations, John Wiley Sons, New York, 1977. 

Evidence of the appHcation of computers and expert systems to instmmental data interpretation is found in the new discipline of chemometrics (qv) where the relationship between data and information sought is explored as a problem of mathematics and statistics (7—10). One of the most useful insights provided by chemometrics is the realization that a cluster of measurements of quantities only remotely related to the actual information sought can be used in combination to determine the information desired by inference. Thus, for example, a combination of viscosity, boiling point, and specific gravity data can be used to a characterize the chemical composition of a mixture of solvents (11). The complexity of such a procedure is accommodated by performing a multivariate data analysis. 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

A general comment that affects all statistical multivariate data analysis techniques, namely that each of the variables should be given equal chance to influence the outcome of the analysis. This can be achieved by scaling the variables in appropriative way. One popular method for scaling variables is autoscaling, whereby the variance of each variable is adjusted to 1. 

In all statistical modeling, including multivariate data analysis, it is crucially important to determine the applicability of the derived model, and for several reasons ... 

This is for univariate data what happens in the case of multivariate (multiwavelength) spectroscopic analysis. The same thing, only worse. To calculate the effects rigorously and quantitatively is an extremely difficult exercise for the multivariate case, because not only are the errors themselves are involved, but in addition the correlation stmcture of the data exacerbates the effects. Qualitatively we can note that, just as in the univariate case, the presence of error in the absorbance data will bias the coefficient(s) toward zero , to use the formal statistical description. In the multivariate case, however, each coefficient will be biased by different amounts, reflecting the different amounts of noise (or error, more generally) affecting the data at different wavelengths. As mentioned above, these... 

PCA [12, 16] is a multivariate statistics method frequently applied for the analysis of data tables obtained from environmental monitoring studies. It starts from the hypothesis that in the group of original data, there is a set of reduced factors or dominant components (sources of variation) which influence the observed data variance in an important way, and that these factors or components cannot be directly measured (they are hidden factors), since no specific sensors exist for them or, in other words, they cannot be experimentally observed. 

In-Kwon Yeo received the PhD degree in Statistics from University of Wisconsin-Madison in 1997. He joined the Department of Control and Instrumentation Engineering, Kangwon National University as a visiting professor in 2000 and the Division of Mathematics and Statistical Informatics, Chonbuk National University as an assistant professor in Korea. He is currently an associate professor at the Department of Statistics, Sookmyung Women s University. His current research interests include data transformations, multivariate time series analysis and generalized additive models. 

Keil, D. et al., Evaluation of multivariate statistical methods for analysis and modeling of immunotoxicology data, Toxicol. Sci. 51, 245, 1999. 

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